"An inquiry done well can help ensure that British abuses are not repeated and restore Britain’s reputation as an anti-torture champion."

Cameron's ruling Conservative party and its coalition partner the Liberal Democrats came to power in May.

The two parties had long called for an investigation into the torture claims during the previous Labour government.

'Devil in the detail'

The allegations include those by Ethiopian-born Binyam Mohamed, who came to Britain in 1994 seeking asylum. He alleged he was questioned in Morocco in 2002 by people using information that could only have come from the British intelligence service.

Mohamed was arrested in Pakistan in 2002 while trying to return to Britain and spent nearly seven years in US custody or in countries taking part in the US-run rendition programme.

After a lengthy campaign by his supporters, he became the first prisoner to be released from Guantanamo Bay under the administration of Barack Obama, the US president, and returned to Britain in February last year.

Clive Stafford Smith, Mohamed's lawyer, said that the inquiry should be held in public.

"This inquiry is welcome but the devil is in the detail: the idea that this inquiry must be heard in private is misguided and wrong," he said.

"For too long our politicians have confused national security with national embarrassment. This creeping secrecy is the scourge of any open democracy and must be stopped if we are to rebuild trust in our government and our reputation overseas."